I would like to know how much of the recent results on the MMP (due to Hacon, McKernan, Birkar, Cascini, Siu,...) which are usually only stated for varieties over the complex numbers, extend to varieties over arbitrary fields of characteristic zero or to the equivariant case.

I assume that the basic finite generation results hold for any such field, by base extending to an algebraic closure, so I would guess that most results should extend without too much difficulty. The particular questions that I am really interested in are:

1) Given a smooth projective rationally connected variety X over a field k of charaterisitic zero, can we perform a finite sequence of divisorial contractions and flips to obtain a Mori fibre space?

and

2) The equivariant version of 1) for the action of a finite group on X.

2 Answers
2

Both of these cases follow more or less automatically from the Minimal Model Program over an algebraically closed field $\bar k$. This is well, known, see for example the original Mori's paper "Threefolds whose canonical bundles are not numerically effective" or Kollár's paper on 3-folds over $\mathbb R$. Iskovskikh and Manin did both versions for surfaces in the 1970s, and their arguments still apply.

The point is that $K_X$ is invariant under the action of any group $G\subset Aut(X)$ and $Gal(\bar k/k)$. So if $\bar C$ is a curve on $\bar X= X\otimes_k \bar k$ with $K_X . \bar C<0$ then $C= \sum_{g\in G} g.\bar C$ (resp. the sum of the conjugates) also intersects $K_X$ negatively.

So you can work with the $G$-invariant (resp. $Gal$-invariant) part of the Mori cone $NE(X)\subset N_1(X)$. If $R$ is an extremal ray of $NE(X)^G$ then the supporting divisor $D$ can be chosen to be $G$-invariant, and it contracts a face of $NE(X)$ (instead of just a ray).
It is either a divisorial contraction over $k$, or a flipping contraction. In the latter case there is a flip defined over $k$, since it is an appropriate relative canonical model, and every canonical model is automatically $G$-equivariant, resp. $Gal$-equivariant, by its uniqueness.

So you just do the MMP over $k$. Unless $K_X$ is pseudoeffective but not effective, MMP terminates by [BCHM], and you get either a minimal model (with $K_{X_{\rm min}}$ nef) of a Mori-Fano fibration.

Finally, if you started with a variety $X$ such that $\bar X$ is covered by rational curves then $K_X$ is not pseudoeffective.

Thanks. I was a bit worried that the singularities that occur might not be Q-factorial (since one is contracting higher dimensional faces) but I suppose all invariant divisors will still be Q-Cartier. It would have been nice if BCHM had had a section on this, at least for the sake of being able to give a precise reference.
–
ulrichMay 1 '10 at 8:40

2

BCHM is not the one and only paper on MMP, the subject was active for decades. Perhaps someone could remember better references and post them, and then feel free to move the accepted answer mark to that one.
–
VA.May 1 '10 at 13:31

Below you will find some references. I would also add a warning that these questions can become very difficult very soon.

Of course, one can base extend to the algebraic closure, and run the mmp there, but what guarantees that even the first step can be performed such that the resulting objects over the algebraic closure is the base extension of something reasonably related to the original object over the original field?

In particular, take a surface defined over $\mathbb R$ and let $X$ denote the corresponding surface over $\mathbb C$. Now if $X$ contains a $(-1)$-curve that has no real points, then when this curve is contracted, the real points, that is the original surface $X_{\mathbb R}$ does not change at all. That itself is of course not a problem and also one may argue that that $(-1)$-curve has to have a conjugate pair and contracting that results in a surface that should be again defined over $\mathbb R$ (In a different space but with the same set of real points as before). You can imagine that in higher dimensions even more complicated problems arise.

Another issue to keep in mind is that in real algebraic geometry people care about keeping the topology the same. Doing a flip screws that up, so the preferred way to do it is without flips. This seems like an impossible proposition, but Kollár's ingenious series of papers does exactly that. Well, actually what he does is that he performs flips on the complexified space but proved that it happens on the imaginary part and hence it does not mess with the topology of the real points. However, you still need to do it in order to keep the mmp going.

Another issue is the rational connectivity and its relation to Mori fiber spaces.

For this, again, one has to be very careful and state exactly what one wants. What notion of rational connectivity are you using? Are the curves defined over $\mathbb R$ or $\mathbb C$? Do you require $X_{\mathbb R}$ or $X$ (or both) be rationally connected?

To illustrate the difficulties, here is a conjecture of Nash (yes, that Nash): Let $Z$ be a smooth real algebraic variety. Then $Z$ can be realized as the real points of a rational complex algebraic variety.

This, actually, turns out to be false. Kollár calls it the shortest lived conjecture as it was stated in 1954 and disproved by Comessatti around 1914 (I don't remember the exact year). However, even if the statement is false, just the fact that it was made and no one realized for 50 years that it was false should show that these questions are by no means easy. (Comessatti's paper was in Italian and I have no idea how Kollár found it.)

Kollár showed more systematically the possible topology types of manifolds that can satisfy this statement. In particular, Kollár shows that any closed connected 3-manifold occurs as a possibly non-projective real variety birationally equivalent to ${\mathbb P}^3$. In other words the way Nash's conjecture fails is on the verge of the difference between projective and proper again showing that these questions are not easy.